As a motivating example, we consider a engineering design problem under uncertainties. Consider a simple cantilever beam where the Young’s modulus \(E\)  and the length \(L\)  are uncertain, i.e., \(\mathbf {x}=(E,L)\) , and the uncertainties can be described by Normal distributions with \(E \sim \mathcal {N} (\mu _1=69e9,\: \sigma _1^2=11.5e9^2 )\)  and \(L \sim \mathcal {N}(\mu _2=0.45, \: \sigma _2^2 = 0.045^2)\) . To keep it analytically tractable for this motivating example, we assume a trivial function \(\mathbf {y=x}\)  (a random vibration problem considered in Section ). Assuming the two random variables are independent, the Fisher information matrix in this case is a diagonal matrix [1]}:
\( \mathbf {F}(\mu _1,\sigma _1,\mu _2,\sigma _2)=\text{diag}\left(\sigma _1^{-2},2\sigma _1^{-2},\sigma _2^{-2},2\sigma _2^{-2}\right)\) 
